Uniqueness of the Freedman-Townsend Interaction Vertex For Two-Form Gauge Fields

Abstract

Let $B_{\mu \nu }^a$ ($a=1,...,N$) be a system of $N$ free two-form gauge fields, with field strengths $H_{\mu \nu \rho }^a=3{\partial }_{[\mu }{B}_{\nu \rho ]}^a$ and free action $S_0$ equal to $(-1/12)\int d^{n}x{g}_{ab}{H}_{\mu \nu \rho }^a{H}^{b\mu \nu \rho }$ ($n\ge 4$). It is shown that in $n>4$ dimensions, the only consistent local interactions that can be added to the free action are given by functions of the field strength components and their derivatives (and the Chern-Simons forms in $5$ mod $3$ dimensions). These interactions do not modify the gauge invariance $B_{\mu \nu }^a\to B_{\mu \nu }^a+{\partial }_{[\mu }{\Lambda }_{\nu ]}$ of the free theory. By contrast, there exist in $n=4$ dimensions consistent interactions that deform the gauge symmetry of the free theory in a non trivial way. These consistent interactions are uniquely given by the well-known Freedman-Townsend vertex. The method of proof uses the cohomological techniques developed recently in the Yang-Mills context to establish theorems on the structure of renormalized gauge-invariant operators.

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• Renormalization of gauge invariant operators and anomalies in Yang-Mills theory
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